Name the mathematical trick

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Can you name the mathematical trick?

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Used to perform surgery and prove the h-cobordism theorem.
Allows to prove the Nullstellensatz from a weak version.
Used to prove complete reducibility of Lie groups.
Proves isotopy of homeomorphisms of a ball that agree on the boundary.
Proves that knots don't have inverse under knot-sum using an infinite sum of knots.
Proves that for any projective module A over R, there is a free module F with A+F = F.
Used to prove a series expansion for the cotangent function.
Used to prove Godel's incompleteness Theorems
Used as a partial replacement of the Axiom of Choice when the Axiom of Replacement is available.
Used to prove Darboux's Theorem in symplectic geometry.

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